| Description |
1 online resource : illustrations (black and white) |
| Contents |
Cover -- Stochastic Limit Theory: An Introduction for Econometricians -- Copyright -- Dedication -- Contents -- From Preface to the First Edition -- Preface to the Second Edition -- Mathematical Symbols and Abbreviations -- Common Usages -- Part I: Mathematics -- 1: Sets and Numbers -- 1.1 Basic Set Theory -- 1.2 Mappings -- 1.3 Countable Sets -- 1.4 The Real Continuum -- 1.5 Sequences of Sets -- 1.6 Classes of Subsets -- 1.7 Sigma Fields -- 1.8 The Topology of the Real Line -- 2: Limits, Sequences, and Sums -- 2.1 Sequences and Limits -- 2.2 Functions and Continuity -- 2.3 Vector Sequences and Functions -- 2.4 Sequences of Functions -- 2.5 Summability and Order Relations -- 2.6 Inequalities -- 2.7 Regular Variation -- 2.8 Arrays -- 3: Measure -- 3.1 Measure Spaces -- 3.2 The Extension Theorem -- 3.3 Non-measurability -- 3.4 Product Spaces -- 3.5 Measurable Transformations -- 3.6 Borel Functions -- 4: Integration -- 4.1 Construction of the Integral -- 4.2 Properties of the Integral -- 4.3 Product Measure and Multiple Integrals -- 4.4 The Radon-Nikodym Theorem -- 5: Metric Spaces -- 5.1 Spaces -- 5.2 Distances and Metrics -- 5.3 Separability and Completeness -- 5.4 Examples -- 5.5 Mappings on Metric Spaces -- 5.6 Function Spaces -- 6: Topology -- 6.1 Topological Spaces -- 6.2 Countability and Compactness -- 6.3 Separation Properties -- 6.4 Weak Topologies -- 6.5 The Topology of Product Spaces -- 6.6 Embedding and Metrization -- Part II: Probability -- 7: Probability Spaces -- 7.1 Probability Measures -- 7.2 Conditional Probability -- 7.3 Independence -- 7.4 Product Spaces -- 8: Random Variables -- 8.1 Measures on the Line -- 8.2 Distribution Functions -- 8.3 Examples -- 8.4 Multivariate Distributions -- 8.5 Independent Random Variables -- 9: Expectations -- 9.1 Averages and Integrals -- 9.2 Applications -- 9.3 Expectations of Functions of X |
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9.4 Moments -- 9.5 Theorems for the Probabilist's Toolbox -- 9.6 Multivariate Distributions -- 9.7 More Theorems for the Toolbox -- 9.8 Random Variables Depending on a Parameter -- 10: Conditioning -- 10.1 Conditioning in Product Measures -- 10.2 Conditioning on a Sigma Field -- 10.3 Conditional Expectations -- 10.4 Some Theorems on Conditional Expectations -- 10.5 Relationships between Sub- -fields -- 10.6 Conditional Distributions -- 11: Characteristic Functions -- 11.1 The Distribution of Sums of Random Variables -- 11.2 Complex Numbers -- 11.3 The Theory of Characteristic Functions -- 11.4 Examples -- 11.5 Infinite Divisibility -- 11.6 The Inversion Theorem -- 11.7 The Conditional Characteristic Function -- Part III: Theory of Stochastic Processes -- 12: Stochastic Processes -- 12.1 Basic Ideas and Terminology -- 12.2 Convergence of Stochastic Sequences -- 12.3 The Probability Model -- 12.4 The Consistency Theorem -- 12.5 Uniform and Limiting Properties -- 12.6 Uniform Integrability -- 13: Time Series Models -- 13.1 Independence and Stationarity -- 13.2 The Poisson Process -- 13.3 Linear Processes -- 13.4 Random Walks -- 14: Dependence -- 14.1 Shift Transformations -- 14.2 Invariant Events -- 14.3 Ergodicity and Mixing -- 14.4 Sub- -fields and Regularity -- 14.5 Strong and Uniform Mixing -- 15: Mixing -- 15.1 Mixing Sequences of Random Variables -- 15.2 Mixing Inequalities -- 15.3 Mixing in Linear Processes -- 15.4 Sufficient Conditions for Strong and Uniform Mixing -- 16: Martingales -- 16.1 Sequential Conditioning -- 16.2 Extensions of the Martingale Concept -- 16.3 Martingale Convergence -- 16.4 Convergence and the Conditional Variances -- 16.5 Martingale Inequalities -- 17: Mixingales -- 17.1 Definition and Examples -- 17.2 Telescoping Sum Representations -- 17.3 Maximal Inequalities -- 17.4 Uniform Square-Integrability |
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17.5 Autocovariances -- 18: Near-Epoch Dependence -- 18.1 Definitions and Examples -- 18.2 Near-Epoch Dependence and Mixingales -- 18.3 Transformations -- 18.4 Adaptation -- 18.5 Approximability -- 18.6 NED in Volatility -- Part IV: The Law of Large Numbers -- 19: Stochastic Convergence -- 19.1 Almost Sure Convergence -- 19.2 Convergence in Probability -- 19.3 Transformations and Convergence -- 19.4 Convergence in Lp Norm -- 19.5 Examples -- 19.6 Laws of Large Numbers -- 20: Convergence in Lp Norm -- 20.1 Weak Laws by Mean Square Convergence -- 20.2 Almost Sure Convergence by the Method of Subsequences -- 20.3 Truncation Arguments -- 20.4 A Martingale Weak Law -- 20.5 Mixingale Weak Laws -- 20.6 Approximable Processes -- 21: The Strong Law of Large Numbers -- 21.1 Technical Tricks for Proving LLNs -- 21.2 The Case of Independence -- 21.3 Martingale Strong Laws -- 21.4 Conditional Variances and Random Weighting -- 21.5 Strong Laws for Mixingales -- 21.6 NED and Mixing Processes -- 22: Uniform Stochastic Convergence -- 22.1 Stochastic Functions on a Parameter Space -- 22.2 Pointwise and Uniform Convergence -- 22.3 Stochastic Equicontinuity -- 22.4 Generic Uniform Convergence -- 22.5 Uniform Laws of Large Numbers -- Part V: The Central Limit Theorem -- 23: Weak Convergence of Distributions -- 23.1 Basic Concepts -- 23.2 The Skorokhod Representation Theorem -- 23.3 Weak Convergence and Transformations -- 23.4 Convergence of Moments and Characteristic Functions -- 23.5 Criteria for Weak Convergence -- 23.6 Convergence of Random Sums -- 23.7 Stable Distributions -- 24: The Classical Central Limit Theorem -- 24.1 The I.I.D. Case -- 24.2 Independent Heterogeneous Sequences -- 24.3 Feller's Theorem and Asymptotic Negligibility -- 24.4 The Case of Trending Variances -- 24.5 Gaussianity by Other Means -- 24.6 -Stable Convergence |
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25: CLTs for Dependent Processes -- 25.1 A General Convergence Theorem -- 25.2 The Martingale Case -- 25.3 Stationary Ergodic Sequences -- 25.4 The CLT for Mixingales -- 25.5 NED Functions of Mixing Processes -- 26: Extensions and Complements -- 26.1 The CLT with Estimated Normalization -- 26.2 The CLT for Linear Processes -- 26.3 The CLT with Random Norming -- 26.4 The Multivariate CLT -- 26.5 The Delta Method -- 26.6 Law of the Iterated Logarithm -- 26.7 Berry-Esséen Bounds -- Part VI: The Functional Central Limit Theorem -- 27: Measures on Metric Spaces -- 27.1 Separability and Measurability -- 27.2 Measures and Expectations -- 27.3 Function Spaces -- 27.4 The Space C -- 27.5 Measures on C -- 27.6 Wiener Measure -- 28: Stochastic Processes in Continuous Time -- 28.1 Adapted Processes -- 28.2 Diffusions and Martingales -- 28.3 Brownian Motion -- 28.4 Properties of Brownian Motion -- 28.5 Skorokhod Embedding -- 28.6 Processes Derived from Brownian Motion -- 28.7 Independent Increments and Continuity -- 29: Weak Convergence -- 29.1 Weak Convergence in Metric Spaces -- 29.2 Skorokhod's Representation -- 29.3 Metrizing the Space of Measures -- 29.4 Tightness and Convergence -- 29.5 Weak Convergence in C -- 29.6 An FCLT for Martingale Differences -- 29.7 The Multivariate Case -- 30: Càdlàg Functions -- 30.1 The Space D -- 30.2 Metrizing D -- 30.3 Billingsley's Metric -- 30.4 Measures on D -- 30.5 Prokhorov's Metric -- 30.6 Compactness and Tightness in D -- 30.7 Weak Convergence in D -- 31: FCLTs for Dependent Variables -- 31.1 Asymptotic Independence -- 31.2 NED Functions of Mixing Processes 1 -- 31.3 NED Functions of Mixing Processes 2 -- 31.4 Nonstationary Increments -- 31.5 Generalized Partial Sums -- 31.6 The Multivariate Case -- 32: Weak Convergence to Stochastic Integrals -- 32.1 Weak Limit Results for Random Functionals |
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32.2 Stochastic Integrals -- 32.3 Convergence to Stochastic Integrals -- 32.4 Convergence in Probability to -- Bibliography -- Index |
| Summary |
'Stochastic Limit Theory', published in 1994, has become a standard reference in its field. Now reissued in a new edition, offering updated and improved results and an extended range of topics, Davidson surveys asymptotic (large-sample) distribution theory with applications to econometrics, with particular emphasis on the problems of time dependence and heterogeneity.-- Provided by publisher |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from digital title page (Oxford University Press, viewed September 30, 2022) |
| Subject |
Econometrics.
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Limit theorems (Probability theory)
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Stochastic processes.
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Stochastic Processes
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Econometrics
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Limit theorems (Probability theory)
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Stochastic processes
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| Form |
Electronic book
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| ISBN |
9780191927201 |
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0191927201 |
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9780192658807 |
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0192658808 |
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9780192658791 |
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0192658794 |
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